### Monday, October 6, 2014 (3:30 p.m. in Yost 306)

**Title: **Stochastic Modeling of Darwinian Evolution in a Chemostat

**Speaker:** Sylvie Méléard (École polytechnique)

**Abstract: **We are interested in modeling the Darwinian evolution

resulting from the interplay of phenotypic variation and natural

selection through ecological interactions, in the specific scales of

the biological framework of adaptive dynamics. We consider a

multi-resources chemostat model, where the competition between

bacteria results from the sharing of several resources which have

their own dynamics. Starting from a stochastic birth and death process

model, we prove that, when advantageous mutations are rare, the

population behaves on the mutational time scale as a jump process

moving between equilibrium states. An essential technical ingredient

is the study of the long time behavior of a chemostat multi-resources

dynamical system. In the small mutational steps limit this process in

turn gives rise to a differential equation in phenotype space called

canonical equation of adaptive dynamics. From this canonical equation

and still assuming small mutation steps, we prove a rigorous

characterization of the evolutionary branching points.